Let us say, I have a distribution with some unknown parameter and I find the MLE of the parameter. I prove that the MLE is inadmissible using some other estimator. Why is this not in a contradiction to the general theorem on asymptotic optimality of maximum likelihood estimates?
Maximum likelihood estimators are the most efficient consistent estimators, yes. However, that is an asymptotic property. In a finite sample, all bets are off. Indeed, the Wikipedia article on maximum likelihood estimation mentions that MLEs have no optimum properties for finite samples.
You might wind up with a situation where, in a finite sample, a biased and inconsistent estimator has lower risk over the entire parameter space. That is, no matter the true value of the parameter, the biased and inconsistent estimator has lower risk than a maximum likelihood estimator.